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Mirrors > Home > MPE Home > Th. List > ifeq1da | Structured version Visualization version GIF version |
Description: Conditional equality. (Contributed by Jeff Madsen, 2-Sep-2009.) |
Ref | Expression |
---|---|
ifeq1da.1 | ⊢ ((𝜑 ∧ 𝜓) → 𝐴 = 𝐵) |
Ref | Expression |
---|---|
ifeq1da | ⊢ (𝜑 → if(𝜓, 𝐴, 𝐶) = if(𝜓, 𝐵, 𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ifeq1da.1 | . . 3 ⊢ ((𝜑 ∧ 𝜓) → 𝐴 = 𝐵) | |
2 | 1 | ifeq1d 4054 | . 2 ⊢ ((𝜑 ∧ 𝜓) → if(𝜓, 𝐴, 𝐶) = if(𝜓, 𝐵, 𝐶)) |
3 | iffalse 4045 | . . . 4 ⊢ (¬ 𝜓 → if(𝜓, 𝐴, 𝐶) = 𝐶) | |
4 | iffalse 4045 | . . . 4 ⊢ (¬ 𝜓 → if(𝜓, 𝐵, 𝐶) = 𝐶) | |
5 | 3, 4 | eqtr4d 2647 | . . 3 ⊢ (¬ 𝜓 → if(𝜓, 𝐴, 𝐶) = if(𝜓, 𝐵, 𝐶)) |
6 | 5 | adantl 481 | . 2 ⊢ ((𝜑 ∧ ¬ 𝜓) → if(𝜓, 𝐴, 𝐶) = if(𝜓, 𝐵, 𝐶)) |
7 | 2, 6 | pm2.61dan 828 | 1 ⊢ (𝜑 → if(𝜓, 𝐴, 𝐶) = if(𝜓, 𝐵, 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 383 = wceq 1475 ifcif 4036 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-rab 2905 df-v 3175 df-un 3545 df-if 4037 |
This theorem is referenced by: ifeq12da 4068 cantnflem1d 8468 cantnflem1 8469 dfac12lem1 8848 xrmaxeq 11884 xrmineq 11885 rexmul 11973 max0add 13898 sumeq2ii 14271 fsumser 14308 ramcl 15571 dmdprdsplitlem 18259 coe1pwmul 19470 scmatscmiddistr 20133 mulmarep1gsum1 20198 maducoeval2 20265 madugsum 20268 madurid 20269 ptcld 21226 copco 22626 ibllem 23337 itgvallem3 23358 iblposlem 23364 iblss2 23378 iblmulc2 23403 cnplimc 23457 limcco 23463 dvexp3 23545 dchrinvcl 24778 lgsval2lem 24832 lgsval4lem 24833 lgsneg 24846 lgsmod 24848 lgsdilem2 24858 rpvmasum2 25001 mrsubrn 30664 ftc1anclem6 32660 ftc1anclem8 32662 |
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